Numeration and discrete dynamical systems

This survey aims at giving both a dynamical and computer arithmetic-oriented presentation of several classical numeration systems, by focusing on the discrete dynamical systems that underly them: this provides simple algorithmic generation processes, information on the statistics of digits, on the mean behavior, and also on periodic expansions (whose study is motivated, among other things, by finite machine simulations). We consider numeration systems in a broad sense, that is, representation systems of numbers that also include continued fraction expansions. These numeration systems might be positional or not, provide unique expansions or be redundant. Special attention will be payed to β-numeration (one expands a positive real number with respect to the base β > 1), to continued fractions, and to their Lyapounov exponents. In particular, we will compare both representation systems with respect to the number of significant digits required to go from one type of expansion to the other one, through the discussion of extensions of Lochs’ theorem.     

Selective carrier removal using oxygen implantation in GaAs

It has been reported that free-carrier compensation can be achieved using oxygen implantation and can remain stable even after annealing at temperatures up to 900°C. It is demonstrated that carrier removal rate is drastically dependent on the initial donor species. It is very likely that carrier removal in Si-doped layers takes place via formation of Si-O complexes.     

Electrical stress and failure mechanism of the winding insulationin PWM-inverter-fed low-voltage induction motors

The winding insulation of low-voltage induction motors in adjustable-speed drive systems with voltage-fed inverters is substantially more stressed than in line-powered motors. Consequently, this operation is subject to limitations depending on the electrical stress and on the failure behavior of the winding insulation. Actual recommendations do not consider sufficiently the physics behind these phenomena and contain large utilizable reserves     

Vibrational properties of the LiNbO3 z-surfaces

The existence of localized vibrational modes both at the positive and at the negative LiNbO3 (0001) surface is demonstrated by means of first-principles calculations and Raman spectroscopy measurements. First, the phonon modes of the crystal bulk and of the (0001) surface are calculated within the density functional theory. In a second step, the Raman spectra of LiNbO(3) bulk and of the two surfaces are measured. The phonon modes localized at the two surfaces are found to be substantially different, and are also found to differ from the bulk modes. The calculated and measured frequencies are in agreement within the error of the method. Raman spectroscopy is shown to be sensitive to differences between bulk and surface and between positive and negative surface. It represents therefore an alternative method to determine the surface polarity, which does not exploit the pyroelectric or piezoelectric properties of the material.     

Odometers on Regular Languages

Odometers or "adding machines" are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on an arbitrary infinite regular language. In this latter situation, if (A,<) is a totally ordered alphabet, then enumerating the words of a regular language L over A with respect to the induced genealogical ordering gives a one-to-one correspondence between ℕ and L. In this general setting the odometer is not defined on a set of sequences of digits but on a set of pairs of sequences where the first (resp. the second) component of the pair is an infinite word over A (resp. an infinite sequence of states of the minimal automaton of L). We study some properties of the odometer such as continuity, injectivity, surjectivity, minimality, .... We then study some particular cases: we show the equivalence of this new function with the classical odometer built upon a sequence of integers whenever the set of greedy representations of all the integers is a regular language; we also consider substitution numeration systems as well as the connection with β-numerations.     

Role of substrate in electrical properties of GaAs implanted layers

The variation of implantation efficiency in different semi-insulating GaAs materials is shown to be correlated with he purity of the starting substrate, i.e. its Cr concentration and more especially its value of ND?NA (shallow levels). This last value appears to be a key parameter in the electrical properties of implanted layers.     

Using Invariants to Solve the Equivalence Problem for Ordinary Differential Equations

Two given ordinary differential equations (ODEs) are called equivalent if one can be transformed into the other by a change of variables. The equivalence problem consists of two parts: deciding equivalence and determining a transformation that connects the ODEs. Our motivation for considering this problem is to translate a known solution of an ODE to solutions of ODEs which are equivalent to it, thus allowing a systematic use of collections of solved ODEs. In general, the equivalence problem is considered to be solved when a complete set of invariants has been found. In practice, using invariants to solve the equivalence problem for a given class of ODEs may require substantial computational effort. Using Tresse's invariants for second order ODEs as a starting point, we present an algorithmic method to solve the equivalence problem for the case of no or one symmetry. The method may be generalized in principle to a wide range of ODEs for which a complete set of invariants is known. Considering Emden-Fowler Equations as an example, we derive algorithmically equivalence criteria as well as special invariants yielding equivalence transformations. Received: May 26, 2000; revised version: September 6, 2000